As implied from the question, does the sun rotate? If so, do other stars not including the sun also rotate? Would there be any consequences if the sun and other stars didn't rotate? Me and my friends have differing views on this, and would like some clarification. Thanks!

@RafaelReiter: a number infinitesmially close to zero.
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Jerry SchirmerMar 26 '13 at 21:40

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@JerrySchirmer: infinitesimally close to zero? I don't think so. Classically, $\{\vec{0}\}$ (or any singleton set) is a null-subset of the set of angular momentum vectors, so one might say the probability is in fact exactly zero. Quantum-mechanically, there are only finitely many states of quantised angular momentum when the energy is limited, so the probability is ridiculously, but not infitesimally small.
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leftaroundaboutMar 26 '13 at 23:38

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@leftaroundabout: Classically, you would interpret the probability of a singleton in the continuum as an infinitesimal number, at least if you're working in a set that allows for infinite and infinitesimal numbers. You know exactly what I mean, though.
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Jerry SchirmerMar 27 '13 at 2:20

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One can see the rotation of the sun by looking at the motion of sunspots on its surface. Over time, the sunspots will move across the sun's surface - proof of its rotation. Furthermore, the rate of the sun's rotation is not constant throughout the sun; it is higher near the equator and slower near the poles.

Other stars rotate as well. To imagine why this would be requires some thought about the creation of a star. A star begins as an enormous cloud of dust and gas. When these clouds form, they always have some rotation - even if this rotation is incredibly small and imperceptible. Gravity, however, begins pulling the cloud together into a smaller and more compact object (a star). The shrinking of the large cloud into a smaller body hugely decreases its moment of inertia, causing its angular velocity to significantly increase by the conservation of angular momentum. (This is much like how a figure skater increases her rate of spinning by pulling in her arms.) Because of this, even the slightest hint of rotation of the large gas cloud is amplified into a rapid spinning once the compact star forms.

Rotation is notable in pulsars, which are rapidly rotating neutron stars. Rapidly spinning neutron stars produce magnetic fields, causing electromagnetic radiation (often in the form of X-rays). Beams of the radiation can strike Earth, allowing observatories to observe the rapidly pulsating stars.

Note: if there was no rotation at all of the cloud of dust, then the planets wouldn't have cross-radial velocity neither, collapsing into the sun or not forming at all. The fact that you can ask the question is proof that the Sun rotates. About other stars, read answer above, it's pretty good.
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Francisco PresenciaMar 27 '13 at 5:30

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It would be an incredibly unlikely occurrence for the protostellar cloud to have so little net angular momentum that, after collapsing (think of the figure skater) it would not have a noticeable rotation. Possible, but ridiculously improbable.
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Larry GritzMar 27 '13 at 19:52

Since we know the characteristic spectrum of light from a hot body of a given temperature, we can use the same effect to determine if other stars rotate as well. Note that this only gives the spread in velocities along the line of sight, so a star may be rotating much more than the amount measured.

So, like, dudes, how should I tan so I get both sides of my face the same?
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KazMar 27 '13 at 1:08

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@BradMace Presumably you're referring to the popular notion that "all motion is relative", which doesn't apply to rotation. The Sun certainly is rotating relative to us, but that just means that it is rotating "relative to itself" as well. Physics in a rotating frame of reference is measurably different from in a non-rotating frame.
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Mark MitchisonMar 27 '13 at 4:15

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@BradMace But the period of the Earth's rotation around the Sun (a year) is well-known by other means :) Also, the Doppler shift shows how one side of the sun is moving towards an approximately inertial reference frame (the Earth), while the other side is moving away from it, hence the Sun is rotating around its own axis. Because the Earth is so far away from the Sun, the apparent motion of the Sun's surface due to the Earth's orbit is approximately perpendicular to the line pointing from the Sun to the Earth, which wouldn't have an effect on Doppler measurements.
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Mark MitchisonMar 27 '13 at 16:06

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This is getting kind of ridiculously long, so I started a discussion in chat so we can all talk about the sun there.
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DanMar 29 '13 at 19:43

Yes, it does rotate. There's an evidence similar to the sunspots. It's not much historical. Yet, we can observe it -The 2012 Venus transit. I noticed this in three of my images (1, 2, 3) which I got during the transit. These are the images from NASA's SDO, captured in the visible spectrum. On viewing these images consecutively, you can clearly see sun's rotation. Or you can also download their amazing video based on the same time-lapse. If you observe it closely, you can see the sun's rotation in the background quite relative to Venus.

Do other stars not including the sun also rotate?

Though Dan's answer provide an excellent evidence, Draksis' answer gives an amazing explanation. As far as we've observed, almost every celestial body in the observable universe rotates about its axis, due to the conservation of angular momentum $mr^2\omega$. As the distance (radius of orbital motion) decreases, $\omega$ should increase. Inertia does this automatically. Interstellar clouds (a few or few thousand LYs long) are massive enough to experience a gravitational collapse. Once this collapse leads to proto-star phase, the center is dense enough, so that the surrounding matter can swirl and finally spiral into it. As the lengthy cloud slowly spiral inwards it spins faster. Hence, almost all stellar objects spin.

Would there be any consequences if the sun and other stars didn't rotate?

Let's assume that the sun and other stars along with the planets suddenly stopped rotating themselves. There won't be a consequence or any effects now. (This necessarily doesn't happen due to conservation of angular momentum). But as Frank says (which according to the currently accepted hypothesis), if the clouds had stopped rotating (no initial angular momentum), there would be no proto-planetary disks surrounding the nebula-forming region, leading to a central non-rotating star with no planets. These proto-stellar disks are the accretion disks (dense arms of the clouds) which sometimes acquire enough density to form a star-like object (unable to fuse further) ending up as a planet.

The photosphere of the Sun rotates with a 25 day period at equator and more than a 36 day period at the poles. Below the photosphere we have the convection zone and below the convection zone everything rotate as it was one solid spherical body. This spherical body is 70 % of the volume of the Sun and even more of the mass and rotates with a period of 26,3 days. The surface of this sphere is called the tachocline and its extremely thin, less than 3% of the solar diameter.

Often magnetic confinement are used as a explanation of how the interior of the Sun can behave like one solid rotating spherical body, and the magnetic field are again hypothesized to come from motion in the convection zone, but this theory is not yet developed to level where it can fully explain what we observe.

Alternatively the interior of the sun is a fluid or solid body, but this conflicts with current fusion theory where the core of the sun is a hot, high pressure, nuclear fusion furnace.

Yes the rotation of stars have some implications, like neutron stars that rotate faster than a dentists drill. If they where made of ordinary matter the centrifugal forces would rip them apart, so we postulated that they where made of neutrons with huge gravity forces holding them together.
Our Sun is the most spherical object we know of and we would excpect it to be like a compressed oval ball due to centrifugal forces, but current theory states that it is the magnetic field which confines the hot plasma ball into the perfect sphere we observe.